Which social decisions are intuitive? Which are deliberative? The dual-process approach to human sociality has emerged in the last decades as a vibrant and exciting area of research. Here, I review the existing literature on the cognitive basis of cooperation, altruism, honesty, equity-efficiency, positive and negative reciprocity, and moral judgments. For each of these domains, I list a number of open problems that I believe to be crucial to further advance our understanding of human social cognition. I conclude by making an attempt to introduce a game-theoretical framework to organize the existing empirical evidence. This framework seems promising, as it turns out to make predictions that are generally in line with the experimental data in all but one domain: positive reciprocity. I tried to keep the review selfcontained, exhaustive, and research-oriented. My hope is that it can contribute to bring further attention to this fascinating area of research.